Frm erspectivities t erspective Cllineatins Given distinct planes π and π 0 in extended Euclidean space E3, and a pint n neither plane, the assciated ne-pint prjectin with center maps π bijectively nt π 0. That s a simple fact, but frm the standpint f real-wrld ne-pint prjectins such as perspective drawing r phtgraphy, this map, called a perspectivity, is actually a bit strange. T see why, we first review three types f practical, ne-pint prjectins in Figure 1. bject Center (sun) Center bject Image bject Image Image Abve left: Illustratin f a camera bscura frm Sketchbk n military art, including gemetry, frtificatins, and pyrtechnics. Seventeenth century, rigin unknwn, pssibly Italian. Center Abve: Hubble Telescpe prtrait f Jupiter and I, July 24, 1996. J. Spencer (Lwell bservatry) and NASA/ESA. Left: Illustratin f the perspective drawing prcess by Albrecht Dürer. Wdcut, 1525. Figure 1: Three practical ne-pint prjectins. In the upper left f Figure 1 is a diagram f a camera bscura, r pinhle camera. A light ray frm an bject (the spire f a dmed building) passes thrugh the center f prjectin (a hle in a wall f a darkened rm) and meets the inverted image n the ppsite wall f the rm. The pht itinerary f the light ray is thus bject-center-image. In the upper right f the figure, 1
a light ray leaves the center f prjectin (the sun), passes by the bject (Jupiter s mn I), and meets its image (the shadw f I n Jupiter). The shadw itinerary f the light ray is thus center-bject-image. Finally, in the lwer left f the figure, a light ray leaves an bject (the mdel s hand), passes thrugh the image (a perspective drawing f the hand), and meets the center f prjectin (the artist s eye). The perspective drawing itinerary is thus bject-image-center. Figure 2 shws that nt ne, but all three types f practical ne-pint prjectins are invlved in a plane-t-plane perspectivity. π INHLE HTGRAH F CEAN IMAGE F THE FAR DISTANT, EAST-SUTHEASTERN CEAN IMAGE F THE FAR DISTANT, WEST-NRTHWESTERN RAIRIE ERSECTIVE DRAWING F RAIRIE RAIRIE SHADW F EACH EACH West CEAN East π Figure 2: Schematic f a plane-t-plane perspectivity. T highlight the different types f practical prjectins invlved in the perspectivity, we divide the hrizntal plane π int three regins: tw infinite half-planes called the cean and the prairie, and a narrw strip dividing them called the beach. The center f prjectin is directly abve the shreline the bundary between the cean and the beach. The perspectivity prjects the plane π nt the vertical plane π, which meets π in the bundary between the cean and the prairie. The entire plane π is paved with an avenue f initials G fr perspective gemetry. Lking at the dashed line thrugh cnnecting the cean and its image, we see that the itinerary f the assciated light ray is bject-center-image: the pht type. Thus the image f the cean is like a pinhle phtgraph: the initials G are reversed, the cean almst beneath is mapped very high up n π, and the far eastern cean is mapped just abve the hrizn line 2
separating the images f the cean and the prairie. Frm the dashed line thrugh cnnecting the beach and its image, we see that the itinerary f the assciated light ray is center-bject-image the shadw type. Thus the image f the beach is essentially an infinitely elngated shadw. Finally, frm the dashed line thrugh cnnecting the prairie and its image, we see that the itinerary f the assciated light ray is bject-image-center the perspective drawing type. Thus the image f the prairie is a perspective drawing n the vertical plane π. The infinite rad f G initials cnverges t a vanishing pint n the hrizn. The strangest and mst interesting part f the image is the neighbrhd f the hrizn line. The hrizn line itself is the image f the line at infinity, r ideal line, in the plane π. Just abve the hrizn lies the image f the far eastern cean, while just belw it lies the image f the far western prairie! Mre generally, as indicated in Figure 2, pints just abve and belw the hrizn cme frm ppsite pints f the cmpass (as applied t an infinite, flat Earth). 3-D letters The exercise f drawing 3-D letters is instructive in relating plane-t-plane perspectivities with perspective cllineatins. Figure 3(a) is a perspective drawing f a 3-D letter Lying n the grund, alng with a pint X which lies in the plane f the upper, gray-shaded face f the L. ur gal is t make a crrect drawing f the pint X which lies in the grund plane directly belw the pint X in space. As a hint, and fr the purpse f ur discussin, Figure 3(b) shws the vanishing pint f the vertical edges f the L, and the vanishing pint f the retreating, hrizntal edges f the L. Finding X given X is like finding A given A, r given ; it exemplifies the basic task invlved in drawing 3-D letters. The pint lies n the hrizn line. The line is the image f the ideal line in E 3 which is incident with bth the upper and lwer planes (frnt and back faces) f the L. X C C X (a) A A (b) Figure 3: Setup f the prblem f cnstructing the image X f X. As is typical when drawing 3-D letters, there are many crrect ways t cnstruct X given X. 3
Figure 4 illustrates a particular methd which will be instructive later. Nte in particular that the figure depicts a plane-t-plane perspectivity in space with center relating the upper and lwer planes f the L; that is, the plane f the gray-shaded surface and the grund plane. The picture plane f the figure that is, the plane f the paper is tilted slightly frward in space, s that we see bth vanishing pints and, which are ideal pints in space that represent perpendicular vertical and hrizntal directins (see als Figure 7). Q k j X X l j Draw the line X until it meets the hrizn line at the pint Q. k Draw the line Q. Since X and Q vanish at Q, they represent parallel hrizntal lines, like the tp and bttm f a fence with a fence pst. l The desired pint X lies at the bttm f the fence, that is, n Q. It als lies n the vertical line X, hence X = X Q. Figure 4: Cnstructin f X given X, reasning in 3-D. There are ther plane-t-plane perspectivities implied by Figure 4; fr example the perspectivity that maps the frnt plane f the letter L nt the plane f the paper, r the perspectivity that maps the back plane f the letter L (the grund plane) nt the plane f the paper. The center f these perspectivities is the reader s viewing eye E (nt pictured in Figure 4, but see Figure 7), assuming it is lcated at the crrect viewpint with respect t the page. These perspectivites are like the ne in Figure 2 with center. Since these mappings are nt the plane f the paper, any pint n the page (say X) can represent a pint in the plane f the gray frnt surface f the L, and likewise any pint n the page (say X ) can represent a pint in the grund plane behind the L. Thus we can think f the mapping X X as representing a mapping frm the plane f the paper t itself. This mapping is a perspective cllineatin. Hwever, because this perspective cllineatin invlves the entire plane f the paper, there are sme pint mappings f the type X X frm the frnt plane f the L t the back plane f the L that are a bit mre challenging t the artist s intuitin methd emplyed in Figure 4. Fr example, Figure 5(a) challenges us t find when the pint is given abve the hrizn line. The figure whimsically depicts this pint as being in the sky f the picture. Recall frm Figure 2 that a cmplete plane-t-plane perspectivity invlves shadw prjectin and perspective drawing and a reversed phtgraphic negative. In particular, the pint in Figure 5(a) is like the pints in the sky f the plane π in Figure 2 which belng t the reversed phtgraphic image f the cean part f the plane π. This makes 3-D intuitin harder t apply when finding in Figure 5(a). 4
Rather than try t draw r imagine all three planes invlved (the plane f the page and the frnt and back planes f the L) it s easier t just wrk with the plane f the page and sme basic prperties f perspective cllineatins. (Hwever, we discuss the three-planes pint f view in the next sectin.) Let s jt dwn what we have t wrk with in Figure 5(b). First, we have the pint whse image we wish t determine. We draw that in Figure 5(b). Secnd, a perspective cllineatin has a unique line f fixed pints called the axis. In Figure 5(a) the hrizn line is the image f the line at infinity that belngs t the (parallel) frnt and back planes f the L, hence the pints f are fixed. Thus in Figure 5(b) is the axis f the perspective cllineatin. Third, a perspective cllineatin has a special fixed pint called the center. Every pint and its image (such as A A,, and C C in Figure 3(b)) are cllinear with the center, hence the pint is the center, as drawn in Figure 5(b). Furth, a perspective cllineatin is cmpletely determined by its center, its axis, and the image f any nn-fixed pint (such as in Figure 5(b)). That s because a perspective cllineatin maps pints t pints, lines t lines, and preserves intersectins. Nw we re ready t slve fr. (a) (b) Figure 5: Tw ways f envisining the prblem. Figure 6 shws hw we can use the abve prperties t find. Cmparisn with the steps in Figure 4 shws that the tw methds are analgus, but the intuitive reasning is different. In Figure 4 we reasned artistically in terms f spatial visualizatin, while the methd f Figure 6 depends strictly upn the prperties f perspective cllineatins. There is n need t think in terms f real bjects in space, but the cnnectin still remains, as we illustrate in the next sectin. 5
j Draw the line ; it meets the hrizn line at the pint R. R k Draw the line R. Since R is fixed (it lies n the axis), the line maps t the line R, hence lies n R. l ut als lies n, hence = R. k j l Figure 6: Cnstructin f given, using the prperties f perspective cllineatins. Thinking in 3-D We can interpret the slutin steps in Figure 6 in terms f 3-D letters in space. Figure 7 illustrates this way f lking at the prblem. A yung reader lks at a versin f Figure 5, which lies in a slanted plane α. The reader s viewing eye is at the pint E (pints and lines utside α are dented with bldface letters). The 3-D letter L lies n a hrizntal plane λ, and the upper (frnt) face f the L lies in the hrizntal plane λ. The ideal pint belngs t all vertical lines, such as the line cnnecting the vertices and f the L. The ideal line belngs t the hrizntal planes λ and λ. The viewer sees the ideal pint as the pint in α, and viewer sees the ideal line as the hrizn line in α. The viewer sees the pint in Figure 5(a) as the image f an ideal pint n, and the viewer sees the pints and as the respective pints and in α. efre discussing the pints and it will be helpful t intrduce sme functin ntatin. We dente three perspectivities as fllws, ne with center and tw with center E: In particular we have : λ λ, E : λ α, Q E : λ α. () =, E () =, Q E ( ) =. We are after the perspective cllineatin : α α, where = Q E 1 E, 6
and in particular we want t find = ( ) = Q E 1 E ( ). Nw as shwn in Figure 7, 1 E ( ) =, where λ and, E, and are cllinear. Similarly, () =, where λ and, and are cllinear. Finally, Q E ( ) =, where α and,, and E are cllinear. reader s eye E α (plane f Figure 5) t ideal pint and ideal line t ideal pint t ideal pint t ideal pint λ λ Figure 7: Visualizing the three planes in space. In Figure 7 we can think f a light ray (dashed line) leaving the vertex f the L, piercing the plane α in the pint, and striking the reader s eye E. This is the bject-image-center itinerary f a perspective drawing setup, like the ne in the lwer left f Figure 1. In this case it is reasnable t think f E as the viewing eye f an nlker. n the ther hand, Figure 7 shws a light ray leaving the pint, passing thrugh the back f the reader s eye E, cming ut the pupil, and striking the plane α at the pint! bviusly the characterizatin f E as a persn s eye is nt apprpriate in this case. In fact, this is the bject-center-image itinerary f a pinhle camera, like the picture in the upper left f Figure 1. Anther drawback f Figure 7 is that it is a rather cmplicated methd f lcating given, by cmparisn with the simple methd f Figure 6. Nevertheless, the idea f Figure 7 suggests an interesting interpretatin f the perspective cllineatin, and it reminds us that even simple exercises like drawing 3-D letters can have deeper implicatins. 7